Best Known (90−16, 90, s)-Nets in Base 9
(90−16, 90, 66433)-Net over F9 — Constructive and digital
Digital (74, 90, 66433)-net over F9, using
- 91 times duplication [i] based on digital (73, 89, 66433)-net over F9, using
- net defined by OOA [i] based on linear OOA(989, 66433, F9, 16, 16) (dual of [(66433, 16), 1062839, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(989, 531464, F9, 16) (dual of [531464, 531375, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(989, 531469, F9, 16) (dual of [531469, 531380, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(985, 531441, F9, 16) (dual of [531441, 531356, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(94, 28, F9, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,9)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(989, 531469, F9, 16) (dual of [531469, 531380, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(989, 531464, F9, 16) (dual of [531464, 531375, 17]-code), using
- net defined by OOA [i] based on linear OOA(989, 66433, F9, 16, 16) (dual of [(66433, 16), 1062839, 17]-NRT-code), using
(90−16, 90, 531471)-Net over F9 — Digital
Digital (74, 90, 531471)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(990, 531471, F9, 16) (dual of [531471, 531381, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(985, 531441, F9, 16) (dual of [531441, 531356, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(955, 531441, F9, 11) (dual of [531441, 531386, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(94, 29, F9, 3) (dual of [29, 25, 4]-code or 29-cap in PG(3,9)), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
(90−16, 90, large)-Net in Base 9 — Upper bound on s
There is no (74, 90, large)-net in base 9, because
- 14 times m-reduction [i] would yield (74, 76, large)-net in base 9, but